Lec 24 – Leibnitz Theorem for nth Derivatives of the Product | MASC-01 Elementary Calculus

Описание: Lecture 24 – Leibnitz Theorem for nth Derivative of a Product | MASC-01 Elementary Calculus
Topic: Leibnitz Theorem (Statement + Proof + Examples)

In this lecture, Dr. Ravi Dwivedi explains the Leibnitz Theorem, which gives a general formula for finding the nth derivative of the product of two functions.
इस वीडियो में हम गुणन के nवें अवकलज के लिए लेइबनिट्ज प्रमेय को आसान भाषा में समझते हैं, उसका पूरा प्रमाण (proof) लिखते हैं, और कई महत्वपूर्ण उदाहरण solve करते हैं।

🔍 📌 What is Leibnitz Theorem? (Easy Wording so YouTube Shows Correctly)

If
y = u(x) · v(x)
then the nth derivative is given by:

y(n) = Σ (from k = 0 to n) [ C(n, k) · u(n−k) · v(k) ]

where

y(n) = nth derivative of y

u(n−k) = (n−k)th derivative of u

v(k) = kth derivative of v

C(n, k) = n choose k (binomial coefficient)

Hindi में:
लेइबनिट्ज प्रमेय किसी भी दो functions के गुणन का nवें क्रम का अवकलज निकालने का सामान्य नियम देता है।

📝 Topics Covered / इस लेक्चर में कवर किए गए टॉपिक्स
English:

Statement of Leibnitz theorem

Full proof explained step-by-step

Meaning of C(n,k) and derivative patterns

Standard functions:

nth derivative of (x^m · e^x)

nth derivative of (x^m · sin x)

nth derivative of (x^m · cos x)

Exam-oriented solved examples

Shortcut method to simplify the summation

Hindi:

लेइबनिट्ज प्रमेय का कथन

प्रमेय का सरल एवं विस्तृत प्रमाण

C(n,k) (binomial coefficient) का उपयोग

महत्वपूर्ण उदाहरण:

(x^m · e^x) का nth derivative

(x^m · sin x), (x^m · cos x)

परीक्षा में पूछे जाने वाले प्रश्न

Summation को simplify करने के तरीके

🎯 Why This Lecture Is Important?

Very important for BSc/BA Mathematics

Engineering first-year calculus का महत्वपूर्ण हिस्सा

IIT-JAM, CUET-PG, GATE, RPSC, UPHESC में बार-बार पूछा जाता है

Useful in Taylor / Maclaurin series

Fundamental tool for higher order differentiation

📌 Course: MASC-01 Elementary Calculus
📌 Lecture 24 Topic: Leibnitz Theorem (Proof + Examples)
👨‍🏫 Instructor: Dr. Ravi Dwivedi

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#LeibnitzTheorem #NthDerivative #ProductRule #HigherOrderDerivatives
#MASC01 #BScMathematics #EngineeringMaths #mathswithdrravidwivedi